United sprang to life in the second half and seized the three points via a cluster of superb team goals from Anthony Elanga, Mason Greenwood and Marcus Rashford.
But it was only thanks to De Gea's first-half heroics that the Reds managed to reach the interval on level terms.
Brentford started the game superbly, buoyed by a passionate home crowd, and the in-form Spaniard was called on several times to make key saves.
Our no.1 thwarted Brentford midfielder Mathias Jensen in particular, who was put through on goal twice in the opening 45, but several other Brentford players were foiled too.